Given that the quadratic function /(x) = 2x² -12x + 15-a(x+m) +n Find(a) the value of m and of n, hence state a coordinates of minimum point.
1. Given that the quadratic function /(x) = 2x² -12x + 15-a(x+m) +n Find(a) the value of m and of n, hence state a coordinates of minimum point.
first,
for sure a=2, directly, as leading coef, alright?
then:
find the axis of symmetry: x=-12/(-2*2)=3
lha ini ternyata harus sama dengan :
its vertex m = -3; in this form: a(x+m)²+n,
how to find n: easy!
calculate f(3), and we get f(3) = -3
lha yaitu n-nya, iya bener itu,
n=-3,
try asking your teacher, why m=-3 not just 3,
then why n immediately taken from f(3).
I give u bonus:
attached.
2. A (-2,1) and B (6,5) are the opposite ends of the diameter of a circle. Find the coordinates of its centre.
we can find the coordinate of the circle centre by using the mean formula for the x and y
point A (-2,1)
x1 = -2 and y1= 1
point B (6,5)
x2 = 6 and y2 = 5
circle centre coordinate
[tex]x = \frac{x1 + x2}{2} = \frac{ - 2 + 6}{2} = \frac{4}{2} = 2[/tex]
[tex]y = \frac{y1 + y2}{2} = \frac{1 + 5}{2} = \frac{6}{2} = 3[/tex]
so, the centre of the circle is (2,3)
3. find the number of factor 960
960,1,2,480,240,120,60,30,15,3,5960 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5
4. find the greatest common factor of 24 and 48
Jawaban:
24
Penjelasan dengan langkah-langkah:
Jadi kalau dari situ, GCF itu mengambil faktor yang sama, tapi diambil pangkat yang kecil, berbanding terbalik dengan LCM
5. 1. The temperature of the body is 15 °C. What is the temperature in a. Reamer scale ? b. Fahrenheit scale ? c. Kelvin scale ?Pakai solving , given and ask ya
ini udah di jawab oleh guru sekolah saya langsung
6. The curve y = 2x³ + ax + bx + 7 has a stationary point at the point (2, -13). a) Find the value of a and the value of b. b) Find the coordinates of the second stationary point on the curve. C)Determine the nature of the two stationary points. d) Find the coordinates of the point on the curve where the gradient is minimum and state the value of the minimum gradient. *If possible pls send it together with the working way, thank you so much
Jawab:
y = 2x³+ax²+bx+7 ⇒ substitute with (2, -13)
-13 = 2(2)³+a(2)²+b(2)+7 (I assume it's supposed ax², not ax)
-13 = 16+4a+2b+7
4a+2b=-36
b = -18-2a ⇒ [1]
a.
stationary point ⇒ gradient = 0 (y' = 0)
y = 2x³+ax²+bx+7
y' = 6x² + 2ax +b
0 = 6x² + 2ax +b ⇒ x = 2
0 = 6(2)² + 2a(2) +b ⇒ substitute with [1]
0 = 24 + 4a + (-18-2a)
2a = -6
a = -3
b = -18-2a = -18-2(-3) = -12
b.
y' = 6x² + 2ax +b
0 = 6x² + 2(-3)x +(-12)
0 = 6x² - 6x -12
0 = x² - x - 2
(x-2)(x+1) = 0
x₁ = 2
x₂ = -1
y = 2x³-3x²-12x+7
y(-1) = 2(-1)³-3(-1)²-12(-1)+7
= 14
coordinate (-1, 14)
c. (-1, 14) maximum point
(2, -13) minimum point
d.
y' = gradient
gradient min = y' min
= 6x² - 6x -12 min ⇒ y'' = 0
y'' = 12x - 6
0 = 12x - 6
x = 1/2
y(1/2) = 2(1/2)³-3(1/2)²-12(1/2)+7
= 1/2
coordinate ([tex]\frac{1}{2}[/tex] , [tex]\frac{1}{2}[/tex])
7. A temple is drawn using a scale of 1: 500. The height of the temple on the drawing is 7 cm. Find the actual height of the temple in meters.
Answer:
35 metresExplanation and steps to answer:
Given that:
Scale (S) = 1 : 500
Height on the drawing (Dh) = 7 cm
Find the actual height (Ah)!
Use this equation:
Ah = Dh/SAh = 7 / (1/500)
Ah = 7 × 500
Ah = 3,500 cm
convert it into m
3,500 cm = 3,500/100 m =
35 metresof actual height.
_____________
#Genius - kexcvi
8. what are the coordinates of Q?
Jawaban:
The x coordinate is 0 at point Q and the y coordinate is 3 at point Q
Penjelasan dengan langkah-langkah:
moga membantu
9. Let y+2x=3 be the equation of the line which meets the curve 2xsquare -3xy=14. Find the coordinates of the points of intersection of the line and the curve.
the intersect points are (-7/8 , 19/4) and (2,-1)
10. Find the coordinates of the points on the curve y=x^3 where the gradient is 12
Penjelasan dengan langkah-langkah:
y' = 3x²
12 = 3x²
x² = 4
x = ±√4
= ± 2
x = 2 maka y= 8
x= -2 maka y = -8
11. Point P has coordinates (k,-3). Point Q has coordinates (2,-3). The length of PQ is 6.5 units and k < 0 Find the value of k.
The value of k is –4.5.
ExplanationDistance of Two Points
It is given that:
Point P has coordinates (k, –3).Point Q has coordinates (2, –3).The length of PQ is 6.5 units.k < 0Question: Find the value of k.
SIMPLER METHOD
Looking at both ordinates, we can get that point P and point Q are collinear, on the line of y = –3. It means the distance between point P and point Q is determined by the difference of their abscissas. Considering that k < 0, we get:
[tex]\begin{aligned}d_{PQ}&=x_Q-x_P\\\Rightarrow 6.5&=2-k\\k&=2-6.5\\\therefore\ k&=\bf{-}4.5\end{aligned}[/tex]
DISTANCE FORMULA METHOD
To find the value of k, we can also use the distance of two points formula derived from the Pythagorean theorem.
[tex]\begin{aligned}&d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\end{aligned}[/tex]
Rewriting that formula based on known facts we have, we get:
[tex]\begin{aligned}d_{PQ}&=\sqrt{\left(x_Q-x_P\right)^2+\left(y_Q-y_P\right)^2}\\\Rightarrow\;{d_{PQ}}^2&=\left(x_Q-x_P\right)^2+\left(y_Q-y_P\right)^2\\\Rightarrow {(6.5)}^2&={(2-k)}^2+\left(-3-(-3)\right)^2\\&={(2-k)}^2+0^2\\\Rightarrow\quad\;6.5&={}\pm\sqrt{(2-k)^2}\\&={}\pm(2-k)\\&=\begin{cases}2-k\\k-2\end{cases}\\\Rightarrow\qquad k&=\begin{cases}2-6.5\\6.5+2\end{cases}\\\Rightarrow\qquad k&=\begin{cases}\bf{-}4.5\\\bf8.5\end{cases}\end{aligned}[/tex]
Eliminating positive value of k, we get:
[tex]\therefore\ k=\bf{-}4.5[/tex]
CONCLUSION
∴ The value of k is –4.5.
12. If the coordinates A(-3,1),B(2,-4),C(7,1) is a corner points of square of abcd then the coordinates of D is
D(5,6),but im not sure
13. A street is 400 m long. It is represented by a length 2 cm on a map. Find the scale of the map.
Penjelasan dengan langkah-langkah:
semoga membantu, jangan lupa follow, tekan tombol thank you, dan rate yaa
14. The line 3x + 2y = 6 translated by T(3,-4), then continued dilatation with the center of O and a scale factor of 2. The result of the transformation shadow is....
Jawab:
Penjelasan dengan langkah-langkah:
Penjelasan dengan langkah-langkah:
3x + 2y = 6 ditranslasi T (3, -4)
x' = x + 3 ---> x = x' - 3
y'= y - 4 ---> y = y' + 4
Subsitusikan x dan y ke dalam persamaan.
3x + 2y = 6
3(x' - 3) + 2(y' + 4) = 6
3x' - 9 + 2y' + 8 = 6
3x' + 2y' = 7
Garis 3x + 2y = 7 didilatasi dengan faktor skala 2.
x' = 2x ---> x = 1/2 x'
y' = 2y ---> y = 1/2 y'
Substitusikan ....
3x + 2y = 7
3(1/2 x') + 2(1/2 y') = 7
3/2x' + y' = 7 ....... (x2)
3x' + 2y' = 14 ------> JAWABAN
15. The coordinates of three points are A(1, 1), B(-1, 4) and C(h, k). Given that the gradients of AB, AC and BC are -3a, 3a and a respectively, find the values of h, k and a.
h = 5, k = 7, a = ½
Explanation/Penjelasan
The coordinates of three points are A(1, 1), B(–1, 4) and C(h, k).
Given that the gradients of AB, AC and BC are –3a, 3a and a respectively, we can get
[tex]\begin{aligned}&&m_{\rm AB}&=\frac{y_{\rm B}-y_{\rm A}}{x_{\rm B}-x_{\rm A}}\\\\&\Leftrightarrow&-3a&=\frac{4-1}{-1-1}\\&&&=\frac{3}{-2}\ =\ -3\cdot\frac{1}{2}\\\\&\therefore&a&=\bf\frac{1}{2}\\\\&\therefore&m_{\rm AC}&=3a={\frac{3}{2}}=\frac{y_{\rm C}-y_{\rm A}}{x_{\rm C}-x_{\rm A}}\\&\Leftrightarrow&\frac{3}{2}&=\frac{k-1}{h-1}\\&\Leftrightarrow&\!\!3h-3&=2k-2\quad...(i)\end{aligned}[/tex]
[tex]\begin{aligned}&&m_{\rm BC}&=a={\frac{1}{2}}=\frac{y_{\rm C}-y_{\rm B}}{x_{\rm C}-x_{\rm B}}\\&\Leftrightarrow&\frac{1}{2}&=\frac{k-4}{h-(-1)}\ =\ \frac{k-4}{h+1}\\&\Leftrightarrow&h+1&=2k-8\\&\Leftrightarrow&h&=2k-9\quad...(ii)\end{aligned}[/tex]
[tex]\begin{aligned}&&(ii)\to(i):\\&&\!\!\!\!3(2k-9)-3&=2k-2\\&\Leftrightarrow&6k-27-3&=2k-2\\&\Leftrightarrow&6k-30&=2k-2\\&\Leftrightarrow&6k-2k&=-2+30\\&\Leftrightarrow&4k&=28\\&&\therefore\ k&=\bf7\\\\&&(ii)\implies h&=2(7)-9\\&&&=14-9\\&&\therefore\ h&=\bf5\end{aligned}[/tex]
CONCLUSION/KESIMPULAN
∴ h = 5, k = 7, a = ½
gradient formula :
m = y2 - y1 / x2 - x1
16. The line 2x + 3y = 8 meets the curve 2x^2 + 3y^2 = 110 at two points. Find the coordinates of two points.
Penjelasan dengan langkah-langkah:
the meeting point of two lines mean that the value of x and y is same for both of the functions.
first, we need to change the functions to x = or y =
2x + 3y = 8
2x = 8 -3y
substitute x to the second function.
2x² + 3y² = 110
(8 - 3y)² + 3y² = 110
9y² - 48y + 64 + 3y² = 110
12y² - 48y + 64 = 110
12y² - 48y + 64 - 110 = 0
12y² - 48y - 36 = 0
divide with 12
y² - 4y - 3 = 0
(y - 3) (y - 1) = 0
y = 3 and y = 1
substitute y value to one of the function to find x
y = 3
2x + 3y = 8
2x + 3(3) = 8
2x + 9 = 8
2x = 8 - 9
2x = -1
x = -½
(-½, 3)
y = 1
2x + 3y = 8
2x + 3 = 8
2x = 8 - 3
2x = 5
x = 2½
(2½, 1)
the meeting point are (-½, 3) and (2½, 1)
17. Find The Highest Common Factor of 12 and 18
Jawaban:
jadi jawaban nya adalah 6
18. The figure is not drawn to scale. Find the area of the shaded triangle.
Jawaban:
484
Penjelasan dengan langkah-langkah:
Area of triangle = Base x Height
= 22 x 22
= 484
So, the area of the shaded triangle is 484
19. Given that a map has a representative fraction of 50 000, i.e. it is drawn to a scale of 1:50 000, find thearea on the map which represents an area of 20 km”.
Jawab:40 cm
Penjelasan dengan langkah-langkah:
Map=2.000.000/50.000=40 cm
20. Find the coordinates of the point A' which is the symmetric point of A(3,2) with respect to the line 2x + y - 12 =0 tolong donggg
2x+y-12=0
-y=2x-12
m¹=-2. m¹xm²=-1
-2xm²=-1
m²=-1/-2
m²=1/2
y-y¹=m(x-x¹)
y-2=1/2(x-3)
y=1/2x-3/2+2
2y-x=3.5
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